Optimal. Leaf size=23 \[ x^3+\frac {x^4}{4+e x+\log \left (4-e^x\right )} \]
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Rubi [F] time = 3.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-192 x^2-64 x^3-12 e^2 x^4+e \left (-96 x^3-12 x^4\right )+e^x \left (48 x^2+16 x^3-x^4+3 e^2 x^4+e \left (24 x^3+3 x^4\right )\right )+\left (-96 x^2-16 x^3-24 e x^3+e^x \left (24 x^2+4 x^3+6 e x^3\right )\right ) \log \left (4-e^x\right )+\left (-12 x^2+3 e^x x^2\right ) \log ^2\left (4-e^x\right )}{-64-32 e x-4 e^2 x^2+e^x \left (16+8 e x+e^2 x^2\right )+\left (-32-8 e x+e^x (8+2 e x)\right ) \log \left (4-e^x\right )+\left (-4+e^x\right ) \log ^2\left (4-e^x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (12 e^2 x^2-3 e^{2+x} x^2+64 (3+x)+12 e x (8+x)-3 e^{1+x} x (8+x)-e^x \left (48+16 x-x^2\right )-2 \left (-4+e^x\right ) (12+(2+3 e) x) \log \left (4-e^x\right )-3 \left (-4+e^x\right ) \log ^2\left (4-e^x\right )\right )}{\left (4-e^x\right ) \left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx\\ &=\int \left (-\frac {4 x^4}{\left (-4+e^x\right ) \left (4+e x+\log \left (4-e^x\right )\right )^2}+\frac {x^2 \left (48+16 \left (1+\frac {3 e}{2}\right ) x-(1-3 e (1+e)) x^2+24 \log \left (4-e^x\right )+4 \left (1+\frac {3 e}{2}\right ) x \log \left (4-e^x\right )+3 \log ^2\left (4-e^x\right )\right )}{\left (4+e x+\log \left (4-e^x\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {x^4}{\left (-4+e^x\right ) \left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx\right )+\int \frac {x^2 \left (48+16 \left (1+\frac {3 e}{2}\right ) x-(1-3 e (1+e)) x^2+24 \log \left (4-e^x\right )+4 \left (1+\frac {3 e}{2}\right ) x \log \left (4-e^x\right )+3 \log ^2\left (4-e^x\right )\right )}{\left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {x^4}{\left (-4+e^x\right ) \left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx\right )+\int \left (3 x^2-\frac {(1+e) x^4}{\left (4+e x+\log \left (4-e^x\right )\right )^2}+\frac {4 x^3}{4+e x+\log \left (4-e^x\right )}\right ) \, dx\\ &=x^3-4 \int \frac {x^4}{\left (-4+e^x\right ) \left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx+4 \int \frac {x^3}{4+e x+\log \left (4-e^x\right )} \, dx+(-1-e) \int \frac {x^4}{\left (4+e x+\log \left (4-e^x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 23, normalized size = 1.00 \begin {gather*} x^3+\frac {x^4}{4+e x+\log \left (4-e^x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 42, normalized size = 1.83 \begin {gather*} \frac {x^{4} e + x^{4} + x^{3} \log \left (-e^{x} + 4\right ) + 4 \, x^{3}}{x e + \log \left (-e^{x} + 4\right ) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 42, normalized size = 1.83 \begin {gather*} \frac {x^{4} e + x^{4} + x^{3} \log \left (-e^{x} + 4\right ) + 4 \, x^{3}}{x e + \log \left (-e^{x} + 4\right ) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 24, normalized size = 1.04
method | result | size |
risch | \(\frac {x^{4}}{\ln \left (-{\mathrm e}^{x}+4\right )+4+x \,{\mathrm e}}+x^{3}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 41, normalized size = 1.78 \begin {gather*} \frac {x^{4} {\left (e + 1\right )} + x^{3} \log \left (-e^{x} + 4\right ) + 4 \, x^{3}}{x e + \log \left (-e^{x} + 4\right ) + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 139, normalized size = 6.04 \begin {gather*} \frac {\frac {x^2\,\left (64\,x+x^2\,{\mathrm {e}}^x-3\,x^2\,{\mathrm {e}}^{x+1}+12\,x^2\,\mathrm {e}-16\,x\,{\mathrm {e}}^x\right )}{{\mathrm {e}}^{x+1}-4\,\mathrm {e}+{\mathrm {e}}^x}-\frac {4\,x^3\,\ln \left (4-{\mathrm {e}}^x\right )\,\left ({\mathrm {e}}^x-4\right )}{{\mathrm {e}}^{x+1}-4\,\mathrm {e}+{\mathrm {e}}^x}}{\ln \left (4-{\mathrm {e}}^x\right )+x\,\mathrm {e}+4}+\frac {x^3\,\left (3\,\mathrm {e}+15\right )}{3\,\left (\mathrm {e}+1\right )}-\frac {16\,x^3}{\left ({\mathrm {e}}^x-\frac {4\,\mathrm {e}}{\mathrm {e}+1}\right )\,{\left (\mathrm {e}+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^{4}}{e x + \log {\left (4 - e^{x} \right )} + 4} + x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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